Quantitative imaging with eigenfunctions of the scattering operator

An inverse scattering method that uses eigenfunctions of the scattering operator is described. This approach provides a unified framework that encompasses eigenfunction methods of focusing and quantitative image reconstruction in arbitrary media. Scattered acoustic fields are expressed using a compact, normal operator. The eigenfunctions of this operator correspond to the far-field patterns of source distributions that are directly proportional to the position-dependent contrast of a scattering object. The eigenfunctions also constitute a basis of an operator that is essentially equivalent to the time-reversal operator previously defined by others. Incident wave patterns specified by these eigenfunctions are used in a method that employs products of numerically calculated fields of the eigenfunctions to represent an unknown scattering medium. Analytic reconstruction formulas are derived both for the linearized inverse scattering problem and for the nonlinear problem in which the total acoustic pressure within the medium can be estimated. A modified eigenfunction imaging method that allows efficient reconstructions of large inhomogeneities is also presented. The methods are applied to obtain quantitative images of various scattering objects that span regions large compared to the wavelength of the acoustic illumination. The eigenfunction method is compared to the method of filtered backpropagation implemented by numerical quadrature and to Fourier inversion implemented by fast Fourier transformation. The results show the capability of the eigenfunction method to image objects with large ka. The results show the eigenfunction method is more efficient than the backpropagation method and can also be more efficient than Fourier inversion when the scattering operator has few eigenvalues.